Measuring Angles: Standard Position, Types, Complementary & Supplementary Angles, and Radian Conversion

Measuring Angles

Angles in Standard Position

Angles are fundamental in trigonometry and are measured in degrees (°) or radians. An angle in standard position has its vertex at the origin of the coordinate plane, with its initial side along the positive x-axis. The amount of rotation from the initial side to the terminal side determines the angle's measure.

• Initial Side: The starting position of the angle (along the positive x-axis).

• Terminal Side: The position after rotation.

• Positive Angles: Measured counterclockwise from the initial side.

• Negative Angles: Measured clockwise from the initial side.

Types of Angles:

• Acute Angle: $0^\circ < \theta < 90^\circ$

• Right Angle: $\theta = 90^\circ$

• Obtuse Angle: $90^\circ < \theta < 180^\circ$

• Straight Angle: $\theta = 180^\circ$

Example: Sketching angles such as $-60^\circ$, $150^\circ$, and $-180^\circ$ in standard position involves rotating from the positive x-axis by the specified amount, using the appropriate direction (counterclockwise for positive, clockwise for negative).

Complementary & Supplementary Angles

Introduction to Complementary & Supplementary Angles

Complementary and supplementary angles are pairs of angles with specific sum properties:

• Complementary Angles: Two angles whose measures add up to $90^\circ$.

• Supplementary Angles: Two angles whose measures add up to $180^\circ$.

Example: The complement of $20^\circ$ is $70^\circ$ because $20^\circ + 70^\circ = 90^\circ$. The supplement of $20^\circ$ is $160^\circ$ because $20^\circ + 160^\circ = 180^\circ$.

Note: Complementary and supplementary angles are always considered positive in standard trigonometric problems.

Solving Problems with Complementary & Supplementary Angles

When solving for unknown angles, use the definitions above. In right triangles, the two non-right angles are always complementary. For algebraic expressions, set up equations based on the sum properties.

• Example: If one angle is $x$, its complement is $90^\circ - x$ and its supplement is $180^\circ - x$.

• Example: If $\angle ABC = 2x + 17^\circ$ and $\angle CBD = x + 20^\circ$ are supplementary, then $2x + 17^\circ + x + 20^\circ = 180^\circ$.

Radian Measure and Converting Between Degrees & Radians

Converting Between Degrees & Radians

Radians are another unit for measuring angles, based on the arc length of a circle. One full revolution (circle) is $360^\circ$ or $2\pi$ radians. The conversion formulas are:

• From degrees to radians: $\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180}$

• From radians to degrees: $\theta_{\text{degrees}} = \theta_{\text{radians}} \times \frac{180}{\pi}$

Example: To convert $120^\circ$ to radians: $120^\circ \times \frac{\pi}{180} = \frac{2\pi}{3}$ radians. To convert $\frac{5\pi}{6}$ radians to degrees: $\frac{5\pi}{6} \times \frac{180}{\pi} = 150^\circ$.